Decimals and Logarithms in the Works? And a Dyer’s Hand too?
Taken from Penn Leary’s website, www.baconscipher.com
The following is partly from my book, The Second Cryptographic Shakespeare, copyright 1990.
Let us begin with the name of Simon Stevin. In 1608, a year before the Sonnets, there was published a book with this title-page:
The Art of Tenths,
Teaching how to performe all Computations
whatsoever, by whole Numbers without
Fractions, by the foure Principles of
Common Arithmeticke: namely, Ad -
dition, Subtraction, Multiplication,
Invented by the excellent Mathematician,
Published in English with some additions
by Robert Norton, Gent.
Imprinted at London by S.S. for Hugh
Aspley, and are to be sold at his shop at
Saint Magnus corner. 1608.
Stevin had, in 1582, imprinted a work called La Practique d’ Arithmeti-
que, and then, in 1585, both in Flemish and in French, La Thiende. An earlier, less facile, notation for expressing fractions in tenths was shown in both. In 1608, in DISME, Stevin’s proposal for the adoption of the decimal system was first translated and printed in London, although Stevin still did not employ decimal points. Here is how he recommended his novel way of computing by decimal fractions:
We will speak freely of the great utility of this invention; I say great, much greater than I judge any of you will suspect, and this without at all exalting my own opinion. . .For the astronomer knows the difficult multiplications and divisions which proceed from the progression with degrees, minutes, seconds and thirds. . .the surveyor, he will recognize the great benefit which the world would receive from this science, to avoid. . .the tiresome multiplications in Verges, feet and often inches, which are notably awkward, and often the cause of error. The same of the masters of the mint, merchants and others. . .But the more that these things mentioned are worth while, and the ways to achieve them more laborious, the greater still is this discovery disme, which removes all these difficulties. But how?
It teaches (to tell much in one word) to compute easily, without fractions, all computations which are encountered in the affairs of human beings, in such a way that the four principles of arithmetic which are called addition, subtraction, multiplication and division, are able to achieve this end, causing also similar facility to those who use
the casting-board (jetons). Now if by this means will be gained precious time. . .if by this means labor, annoyance, error, damage, and other accidents commonly joined with these computations be avoided, then I submit this plan voluntarily to your judgment.
Stevin’s ideas caused a revolution in ordinary arithmetic. He recommended converting all of the odd and varying fractions to be found, then and still in the measurement of weights, volume, length, angles and coinage, into tenths or hundredths or thousandths. Such new ways of measuring did not become universal in France until the
metric system was adopted, but the concept has since spread over the world, especially for scientific uses, and has led to far greater efficiency and accuracy in the handling of numbers. Stevin’s tools multiplied the skills of astronomers who were then trying to work from circles to ellipses in their studies of the orbits of the planets. Even some gamblers, at “the casting-board,” benefited. Meanwhile, John Napier had already been practicing those methods.
What did Shakespeare know about Disme and his contribution to mathematics and technology? Read a few
lines from “Troylus and Cressida” (ii, 2, 15):
Surety secure: but modest Doubt is cal’d
The Beacon of the wise: the tent that searches
To’th’bottome of the worst. Let Helen go,
Since the first sword was drawne about this question
Every tythe soule ‘mongst many thousand dismes,
Hath bin as deere as Helen: I meane of ours:
If we have lost so many tenths of ours
To guard a thing not ours, nor worth to vs
(Had it our name) the valew of one ten;
Troy. Fie, fie, my Brother;
Weigh you the worth and honour of a King
(So great as our dead Father) in a Scale
Of common Ounces? Will you with Counters summe
The past proportion of his infinite,
And buckle in a waste most fathomlesse,
With spannes and inches so diminutive,
As feares and reasons? Fie for godly shame?
The author had read Stevin and understood the application of Disme to awkward English inch-pound-gallon measurements, and the need for reform.
We may note, in passing, that Hugh Aspley published Stevin’s book in 1608, William Aspley did the same for the Sonnets in 1609, and W. Aspley was a co-publisher of the 1623 Folio.
About this time (1609) Napier was finishing his Herculean task of the calculation of the logarithmic tables. He had been working on them since 1590, or thereabouts. These tables, when they were published, showed that he had himself made use of decimals and of the period as a separatrix–the decimal point.
The real and worthy object of Francis Bacon’s Dedication to the Sonnets was John Napier. The mathematician from Edinburgh had hugely simplified ordinary calculation (ciphering) by the invention of natural logarithms; he had then redefined for his special purpose the value of unity (the number one) as equal to zero. He had suggested that
principle to Henry Briggs (a co-founder with Francis Bacon of the Virginia Company on Roanoke Island) who then chose an equation for the foundation of logarithms to the base 10. So also had he embraced Stevin’s decimal system. The efficiency of mathematics had thereby been improved by many orders of magnitude.
The thirty superfluous decimal points of the Sonnet Dedication (above right) are Francis Bacon’s tribute to Napier’s accomplishments.The man who inserted them was also well aware of the basis for logarithms; he knew of it before 1609 when the Sonnets were registered and printed, and after Stevin’s book was published in 1608, and knew
of it before the books of Napier and Briggs were published afterward.
Here are a few lines from the verse shown above (left):
In things of great receit with ease we prooue,
Among a number one is reckon’d none.
Then in the number let me passe vntold,
Though in thy stores account I one must be,
Only in a table of logarithms does 1 = 0. The log of 1 is zero, the log of 10 is one, the log of 100 is two, etc.
Logarithms are used mostly “in things of great receipt,” that is, with large numbers to simplify multiplication and division and in calculating powers and roots. But in “thy stores account” (a simple inventory) one still equals one (roman numeral “I”) and must be counted in the conventional manner.
By using logarithms we substitute the simple process of addition for the more involved process of multiplication. Instead of multiplying two numbers together, as 123 X 456 = 56,088. we add their logarithms and look up the corresponding number in a log table:
log 123 + log 456 = 2.08991 + 2.65896 = 4.74887 = log 56,088
This sonnet 136 is mostly meaningless rubbish, except for these four lines. In his acknowledged writings, Francis Bacon was notorious for his use of metaphor, innuendo and ambiguity. Here he offers us a computational puzzle to solve and to celebrate the invention of logarithms.
Unfortunately, there is no other evidence than this of William Shakespeare’s mastery of Elizabethan mathematics. But no doubt the Stratfordians will seize upon this news as proof that Shakespeare attended Cambridge University, perhaps as a classmate of Francis and Anthony Bacon, and that he put such tables of logarithms to good use in figuring the interest on his usurious Stratford loans. This is the traditional resort of academe to the art of “Shakespin.”
On the other hand, some Stratfordians may view this discovery as dangerous. How might Shakespeare have learned advanced mathematics in his little schoolroom at Stratford? To ponder on that, one might be aroused to question the authorship. Better to deny that these four lines have any reference to logarithms. That shall be the
Nevertheless, the inclusion of this timely and knowing reference to logarithms in Sonnet 136 permits us to draw several new and demonstrable conclusions:
This Sonnet was written within a year before the 1609 stated publication date, not years before while it was being “passed about among the poet’s friends.”
Edward De Vere, the Earl of Oxford, whose writings are supposed to have been put aside to ripen for many years, can have had no hand in its composition because he died in 1604.
Christopher Marlowe, likewise, is no longer one of the usual suspects, he having died in 1593.
Francis Bacon explains an original cipher method as follows:
First let all the Letters of the Alphabet , by transposition, be resolved into two Letters onely; for the transposition of two Letters by five placeings will be sufficient for 32. Differences, much more for 24. which is the number of the Alphabet . The example of such an Alphabet is on this wise.
An Example of a Bi-literarie Alphabet.
[A (Aaaaa) ... Z (babbb)]
The excerpt above is exactly copied [scanned], including the periods and their placement. It may be worth noticing that three of the periods are missing, while one is misplaced. The following is a table of the Binary Scale, upon which the calculating ability of modern computers is based:
0 1 2 3 4 5
00000 00001 00010 00011 00100 00101
6 7 8 9 10 11
00110 00111 01000 01001 01010 01011
12 13 14 15 16 17
01100 01101 01110 01111 10000 10001
18 19 20 21 22 23
10010 10011 10100 10101 10110 10111
Charles S. Ingram (who wrote under the name of Jacobite) seems to have been the first to notice the similarity between the Binary Scale and Bacon’s Bi-literarie alphabet;he called attention to it in the English periodical Baconiana (No. 160, March 1960, p. 12). The invention of the Binary Scale traditionally has been credited to Leibniz who devised
a calculating machine in 1671 and found the binary useful for his purposes, though there is evidence that it was known in an earlier century. The binary scale has been extended and continues as the ASCII “code” which is now used in most computers and telecommunication systems.
Therefore, Bacon in an earlier Latin edition of the Advancement of Learning (De Augmentis Scientarium published in 1623) and Leibniz in 1671 produced the same tables; in Bacon’s cipher version “0″ = “a” and “1″ = “b”, and this is imitated in Leibniz’ arithmetical notation. And John Napier, who invented logarithms, had previously illustrated the
use of the binary scale in his Rabdologiae published in 1617.
This is hardly a trivial coincidence. It should be recognized that Francis Bacon had more than a passing interest in basic mathematics, in addition to his known and often published “call for papers” in experimental, observational and empirical scientific research. And, as will be seen, Bacon and John Napier were in communication.
In passing, let us admire a clipping from the web page of one Nigel Davies. Nigel is a Shakespearoid in the strongest sense of Mark Twain’s term. Here goes his version of Sonnet 136:
In things of great receipt with ease we prove
Among a number one is reckoned none.
* The first Will in line 5 refers to both sexual desire and the author’s sexual organ.
* The second will in line 5 augments the first Will so exemplifying the author’s desire to fulfil which is itself augmented in the next line by fill it full..
* The wills in line 6 refers to other lovers’ sexual organs and the will in the same line refers to the author’s sexual organ.
* treasure is Elizabethan slang for a woman’s genitalia.
* Among a number one is reckoned none means that the author being one more who is sexually active with the female subject will not count for anything, i.e. “what difference would another one make?” as well as the fact that, literally, the number 1 was not considered a number.
* one is repeatedly reinforced via one is reckoned none.
* The reference to great receipt suggests that the female subject is in receipt of a great many lovers and the financial terminology in the sonnet (receipt, number, account, reckoned and possibly check) may allude to the woman being a prostitute.
Nigel is not necessarily a sex fiend. He has taken his cue on “Will” from other, previous, Shakspeareoid critics of the Sonnets. For example:
“It is difficult, with any decency, to be more explicit about all those wills — The meaning is not at all difficult to follow, once the veil of humbug is removed from the eyes.”
This is the assessment of A. L. Rowse, Shakespeare’s Sonnets, 1954. The all-seeing professor continues:
“Store’s account” — let’s face it — means simply the number of men the lady had accomodated. There are other plays on words throughout the sonnet, “come,” “things of great receipt,” “hold,” “a something sweet” — drop the Victorian humbug and we may laugh as the Elizabethans laughed. . .from the story we learn two things: that the lady, though no better than she should be, had not yet consented to take the poet; and secondly, the tremendous fact that the name of the author of the Sonnets was Will.
So much for the naughty Prof. Rowse, that scourge of humbug.
We shall proceed with our own discussion of the author’s purposes.
Let us look at all of the critical Sonnets. (The underlining has been added for demonstration purposes):
135 has certain peculiarities; the words “over-plus,” “boot,” “addition,” “addeth” and “adde” are all included.
Then there are 14 examples of “wil” or “will.”
136 has 7 examples of “wil” or “will.” Debauched Stratfordian literary critics of academe have determined that these are dirty words, referring to sexual organs. But we shall chastely ignore the critics and simply add them up. To 21.
What significance has this number? Why, there are 21 letters in Bacon’s truncated key alphabet. Maybe we are now getting a lesson in simple arithmetic, rather than logarithms. Better than sex, anyway. Go
back and look at these two sonnets again:
Whoever hath her wish, thou hast thy Will,
And Will to boot, and Will in overplus;
More than enough am I that vex thee still,
To thy sweet will making addition thus.
Wilt thou, whose will is large and spacious,
Not once vouchsafe to hide my will in thine?
Shall will in others seem right gracious,
And in my will no fair acceptance shine?
The sea all water, yet receives rain still
And in abundance addeth to his store;
So thou, being rich in Will, add to thy Will
One will of mine, to make thy large Will more.
Let no unkind no fair beseechers kill;
Think all but one, and me in that one Will.
If thy soul cheque thee that I come so near,
Swear to thy blind soul that I was thy 'Will,'
And will, thy soul knows, is admitted there;
Thus far for love my love-suit, sweet, fulfil.
'Will' will fulfil the treasure of thy love,
Ay, fill it full with wills, and my will one.
In things of great receipt with ease we prove
Among a number one is reckon'd none:
Then in the number let me pass untold,
Though in thy stores' account I one must be;
For nothing hold me, so it please thee hold
That nothing me, a something sweet to thee:
Make but my name thy love, and love that still,
And then thou lovest me, for my name is 'Will.'
But why is there so much fuss about this word “Will”? There are 14 examples of “W” used as the initial large
capitals of the sonnet verses which are printed as “VV.” Let us consider “Will” as “VVILL.” Could these be Roman
numerals? Let’s check:
The total is 111. Is the poet demanding that we read and interpret sonnet 111? Remember Sonnet 135 and the
words “over-plus,” “boot,” “addition,” “addeth” and “adde”.
O! for my sake do you with Fortune chide,
The guilty goddess of my harmful deeds,
That did not better for my life provide
Than public means which public manners breeds.
Thence comes it that my name receives a brand,
And almost thence my nature is subdued
To what it works in, like the dyer's hand:
Pity me, then, and wish I were renewed;
Whilst, like a willing patient, I will drink
Potions of eisel 'gainst my strong infection;
No bitterness that I will bitter think,
Nor double penance, to correct correction.
Pity me then, dear friend, and I assure ye,
Even that your pity is enough to cure me.
He says his “name receives a brand.” In other words, an indelible mark. His “nature [name] is subdu’d
To what it workes in, like the Dyers hand.”
A Dyers hand takes on a new color, depending on what it works in. So might the poets’s name; it is changed,
but is still recognizable, as is the Dyer’s hand whatever its color. Maybe we are getting somewhere.
Consider: puBlIcK mEaNes=BIKEN
Consider: puBlIcK mAnNers=BIKAN
Following our instructions, what we have found is the author’s name in the most elementary of ciphers: an acrostic of alternate letters. No translation to a trucated key alphabet is needed here, no monoalphabetic substitution is required — all has been simplified for our instruction. And the name is repeatedly misspelled so there can be no mistake as to the intention. In addition, the words “my name” are found on the next line.
The name has received a brand. “BIKEN” and “BIKAN” are homonyms, phonetic spellings of the author’s name, and we can recognize it just as we can the Dyer’s own hand, whether red, green or blue.
This may remind of us of the cipher found in the Sonnet Title Page and Dedication:
o o n y p i r c y p p h r s b e k a a n b a c o n
Bacon was taking great risks here. He put the homonym (BEKAAN) next to his correctly spelled name to illustrate his artifice.
More than 300 years ago it was well known to cryptanalysts that, if there was some reason to suspect that a place or personal name might be included in a monoalphabetic cipher, the cipher could be broken by the “brute force” method. This involved testing the text for 2nd letter, 3rd letter, etc. variations, or even keyed alphabets, and was very slow, but enormously useful. In fact, this was how I found this string of letters.
I quote Giovanni Battista Porta who published, in 1563, a famous cryptographic book, De Furtivis Literarum Notis :
“He urged the use of synonyms in plaintexts, noting that ‘It will also make for difficulty in the interpretation [cryptanalysis] if we avoid the repetition of the same word.’ Like the Argentis, he suggested deliberate misspellings of plaintext words: ‘For it is better for a scribe to be thought ignorant than to pay the penalty for the detection of
plans,’ he wrote.”